1. Algebra 1
Section 4.7

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3. Absolute Value
The absolute value of a number refers to its distance from the origin.
This will help us interpret absolute value inequalities.

4. Absolute Value Inequalities
To find the solutions of |x| < 3, we must find the numbers whose distance from the origin is less than 3.
Examples: -2, 1.5, 0, -0.5

5. Absolute Value Inequalities
This is equivalent to:
x > -3 and x < 3
-3 < x < 3

6. Absolute Value Inequalities
The solution to the absolute value inequality |x| ≥ -3 includes all numbers whose distance from the origin is greater than or equal to 3.

7. Absolute Value Inequalities
This is equivalent to:
x ≤ -3 or x ≥ 3

8. Absolute Value Inequalities
If |x| > c,
then x < -c or x > c
If |x| < c,
then x > -c and x < c

9. Example 1
|x| > 1
x < -1 or x > 1

10. Example 1
|x| ≤ 4
-4 ≤ x ≤ 4

11. Example 2 |x + 5| ≤ 10 x + 5 ≥ -10 and x + 5 ≤ 10 x ≥ -15 and x ≤ 5

12. Example 3 |-3x + 6| > 18 -3x + 6 > 18 or -3x + 6 < -18

13. Absolute Value Inequalities
Be sure the absolute value is isolated before you write the two separate inequalities.

14. Absolute Value Inequalities
Type of Solution
Set Operation
Inequality
conjunction (and)
< or ≤
intersection
disjunction (or)
> or ≥
union

15. Example 4
|t – 98.6| ≤ 2
optimal temperature
within 2°

16. Example 4 |t – 98.6| ≤ 2 t – 98.6 ≤ 2 and t – 98.6 ≥ -2

17. Homework: pp